UTA Department of Mathematics

Algebra Seminar

Date/Time/Room: Friday (09/24/2004) at 4:00pm in 487 Pickard Hall

Speaker: Bill Purpura, Department of Mathematics, University of Texas at Arlington

"Representations of Finite Planes"

Abstract: In this talk, we present two methods of representing finite planes (projective or affine) via algebra. The first is the coordinatization method developed by Marshall Hall in the 1950's. It turns out that the set coordinatizing a finite plane by the method of Hall can be given some algebraic structure. This resulting structure, called a planar ternary ring, is a double loop. Conversely, starting out with any PTR, we can construct a finite plane from it. In this way, we can transform geometrical questions about the plane into algebraic questions about a PTR coordinatizing it. The second representation, due to Andre, applies to translation planes. Translation planes may be viewed either as a splitting normal partition over some group G, or as a spread over some vector space V. These representations happen to be equivalent to each other. As one would expect, a splitting normal partition of a group G, or a spread of a vector space V will yield a translation plane.